Toc

1. Calculus

Calculus

There are two components to calculus. One is the measure the rate of change at any given point on a curve. This rate of change is called the derivative. The simplest example of a rate of change of a function is the slope of a line. We take this one step further to get the rate of change at a point on a line. The other part of calculus is used to measure the exact area under a curve. This is called the integral. If you wanted to find the area of a semicircle, you could use integration to get the answer.

The two parts; the derivative and the integral are inverse functions of each other. That is, they cancel each other out.
Just as (x²)^1/2=x,
the derivative of (integral (x)) = x and
derivative of (integral (f (x)) = f(x).

The derivative is a composite function. This means it is a function acting on another funcion. In fact, the function, is the input instead of just x. The derivative, then takes a type of formula and turns it into another simiilar type of formula. So, a polynomial will always yield a polynomial derivative. A trigonomic function will always yield a trigonomic derivative. There are a few exceptions, but this is generally the case. This is also true for the integral

Geometrically, the derivative can be perceived as the slope of the tangent line to a curve at a given point. This is roughly how steep the curve is at a given point. We can easily find the rate of change of a line just by finding the slope. But, most formulas are not as simple as a line and they're usually curved. We use the basic formula of a line to get the derivative. If you remember the slope of a line is:

y₂-y₁

x₂-x₁

where the change in height is divided by the change in width. The derivative is derived by using this formula and taking x₂to be infinitely close to x₁. When plugging differnt formulas into this formula, they are translated to another similar type formula. The general formula for the derivative is:
The derivative of f(x) = limit as x₂ goes to x₁ of the formula:

f(x₂)-f(x)₁

x₂-x₁

This can be written in shorthand as:

f'(x) = lim f(x₂) - f(x₁)

x₂ -->x₁ x₂ - x₁

So, what would happen if we plug in a formula like x² in the above equation? Remember, we're dealing with a composite funcion, so f(x) is substituted instead of x. So, our derivative of f(x) = x² is:



f'(x) = lim x₂² - x₁²

x₂ --> x₁ x₂ - x₁


= 2x

There is a proof to this, but I don't think that is necessary in this brief introduction to calculus. But, did you notice what happened? (x<SUP2< sup>)' becomes 2x. The more general formula for the derivative of a polynomial term ,with no constant term in front of the x, is
(xⁿ)' = nx^n-1So, if f(x) = x⁴, then f'(x) = (4)x³ = 4x³. And, for any polynomial term with a constant factor as in f(x) = 5x³, the formula is
f'(x) = (3)(5)x² = 15x² where you take n times the constantt. So, the general formula for the derivative for the polynomial is
(cxⁿ)'=(c)(n)x^n-1
where c is any constant.There are other types of derivatives. Some of the main ones are:

If f(x) = e^x, then f'(x) = e^x
If f(x) = sin x, then f'(x) = cos xand if f(x) = sin nx, then f'(x) = n cos nx
If f(x) = cos x, then f'(x) = -sin x andif f(x) = cos nx, then f'(x) = -nsin xThere are many other types, but these are a few of the main ones to get you started.